Joseph Fourier showed that almost any periodic signal can be built from sines and cosines. The Fourier transform reveals which frequencies are present — turning time into frequency.
The continuous form is X(f) = ∫ x(t) e^(−i2πft) dt. In plain language: multiply the signal by a rotating complex exponential at frequency f, integrate, and you learn how much of that frequency exists.
Time vs frequency
A musical chord looks complex in time but simple in frequency — three spikes. An MRI image is built by Fourier-transforming radio signals from hydrogen atoms. JPEG compression removes high frequencies your eyes won't miss.
Where it lives today
MP3 audio, noise-canceling headphones, radio astronomy, image processing, and neural network convolutions all rely on Fourier analysis. It is one of the most deployed equations in engineering.
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